A120112 -id:A120112 - cp's OEIS frontend

A133233 Triangle A133232 read by rows with an additional column T(n,0)=1 added to the left.

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 3, 4, 1, 1, 1, 3, 4, 5, 1, 1, 1, 3, 4, 5, 1, 1, 1, 1, 3, 4, 5, 1, 7, 1, 1, 1, 3, 1, 5, 1, 7, 8, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11

Offset: 0

Mats Granvik, Oct 13 2007

Attaching an additional 1 does not change the composition compared to A133232 since neither the LCM over the elements of a row nor their product is affected.

			The first rows of the triangle and the least common multiple of the rows are:
lcm{1} = 1
lcm{1, 1} = 1
lcm{1, 1, 2} = 2
lcm{1, 1, 2, 3} = 6
lcm{1, 1, 1, 3, 4} = 12
lcm{1, 1, 1, 3, 4, 5} = 60
lcm{1, 1, 1, 3, 4, 5, 1} = 60
lcm{1, 1, 1, 3, 4, 5, 1, 7} = 420
lcm{1, 1, 1, 3, 1, 5, 1, 7, 8} = 840
lcm{1, 1, 1, 1, 1, 5, 1, 7, 8, 9} = 2520
Multiplying the terms in the rows produces the same result:
1 = 1
1*1 = 1
1*1*2 = 2
1*1*2*3 = 6
1*1*1*3*4 = 12
1*1*1*3*4*5 = 60
1*1*1*3*4*5*1 = 60
1*1*1*3*4*5*1*7 = 420
1*1*1*3*1*5*1*7*8 = 840
1*1*1*1*1*5*1*7*8*9 = 2520

Mats Granvik, Table of n, a(n) for n = 0..434

Cf. A003418, A120112, A000961, A014963.

T(n,0) = 1.

T(n,k) = A133232(n,k), k>0.

Removed information which duplicates A133232; offset set to 0 - R. J. Mathar, Nov 23 2010

A133232 Triangle T(n,k) read by rows with a minimum number of prime powers A100994 for which the least common multiple of T(n,1),..,T(n,n) is A003418(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 1, 3, 4, 1, 1, 3, 4, 5, 1, 1, 3, 4, 5, 1, 1, 1, 3, 4, 5, 1, 7, 1, 1, 3, 1, 5, 1, 7, 8, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1, 13, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1

Offset: 1

Mats Granvik, Oct 13 2007

Checked up to 28th row. The rest of the ones in the table are there for the least common multiple to calculate correctly.

			2 occurs 2*1 = 2 times in column 2.
3 occurs 3*2 = 6 times in column 3.
4 occurs 4*1 = 4 times in column 4.
5 occurs 5*4 = 20 times in column 5.
k occurs A133936(k) times in column k. The first rows of the triangle and the least common multiple of the rows are:
lcm{1} = 1
lcm{1, 2} = 2
lcm{1, 2, 3} = 6
lcm{1, 1, 3, 4} = 12
lcm{1, 1, 3, 4, 5} = 60
lcm{1, 1, 3, 4, 5, 1} = 60
lcm{1, 1, 3, 4, 5, 1, 7} = 420
lcm{1, 1, 3, 1, 5, 1, 7, 8} = 840
lcm{1, 1, 1, 1, 5, 1, 7, 8, 9} = 2520

Mats Granvik, Oct 13 2007, Table of n, a(n) for n = 1..406

Cf. A003418, A120112, A014963, A134579.

=if(and(row()>=column();row()A120112));column();1)

=if(and(n>=k; n < A014963*A100994); A100994; 1) - Mats Granvik, Jan 21 2008

A120112 := proc(n) 1-ilcm(seq(i,i=1..n+1))/ilcm(seq(i,i=1..n)) ; end proc:
A133232 := proc(n) if n < k*(1+abs(A120112(k-1))) then k else 1; end if; end proc:
seq(seq(A133232(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Nov 23 2010

b[n_] := b[n] = If[n == 0, 1, LCM @@ Range[n]];
c[n_] := 1 - b[n+1]/b[n];
T[n_, k_] := If[n < k*(1+Abs[c[k-1]]), k, 1];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 01 2021 *)

T(n,k) = if nA120112(k-1)| then k, else 1 (1<=k<=n).

T(n,k) = if n < A014963(k)*A100994(k) then A100994(k), else 1 (1<=k<=n). - Mats Granvik, Jan 21 2008

Indices added to formulas by R. J. Mathar, Nov 23 2010

A120111 Bi-diagonal inverse matrix of A120108.

Original entry on oeis.org

1, -2, 1, 0, -3, 1, 0, 0, -2, 1, 0, 0, 0, -5, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -7, 1, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, -3, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11, 1

Offset: 0

Paul Barry, Jun 09 2006

Subdiagonal is -lcm(1,...,n+2)/lcm(1,...,n+1) or -A014963(n+1).

Row sums are A120112.

			Triangle begins
   1;
  -2,  1;
   0, -3,  1;
   0,  0, -2,  1;
   0,  0,  0, -5,  1;
   0,  0,  0,  0, -1,  1;
   0,  0,  0,  0,  0, -7,  1;
   0,  0,  0,  0,  0,  0, -2,  1;
   0,  0,  0,  0,  0,  0,  0, -3,  1;
   0,  0,  0,  0,  0,  0,  0,  0, -1,   1;
   0,  0,  0,  0,  0,  0,  0,  0,  0, -11, 1;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A014963, A120108, A120112.

A014963:= func< n | Lcm([1..n])/Lcm([1..n-1]) >;
A120111:= func< n,k | k eq n select 1 else k eq n-1 select -A014963(n+1) else 0 >;
[A120111(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, May 05 2023

T[n_, k_] := Switch[k, n, 1, n-1, -Exp[MangoldtLambda[n+1]], _, 0];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* Jean-François Alcover, Mar 01 2021 *)
(* Second program *)
A014963[n_]:= LCM@@Range[n]/(LCM@@Range[n-1]);
A120111[n_, k_]:= If[k==n, 1, If[k==n-1, -A014963[n+1], 0]];
Table[A120111[n,k], {n,0,20}, {k,0,n}]//Flatten (* G. C. Greubel, May 05 2023 *)

def A014963(n): return lcm(range(1,n+1))/lcm(range(1,n))
def A120111(n,k):
    if (kA014963(n+1)
    else: return 1
flatten([[A120111(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, May 05 2023

A276048 Sequence associated with the functional equation of the Riemann zeta zero spectrum (see formulas).

Original entry on oeis.org

0, 2, 9, 2, 625, 1, 117649, 2, 9, 1, 25937424601, 1, 23298085122481, 1, 1, 2, 48661191875666868481, 1, 104127350297911241532841, 1, 1, 1, 907846434775996175406740561329, 1, 625, 1, 9, 1, 88540901833145211536614766025207452637361, 1

Offset: 1

Mats Granvik, Aug 17 2016

nonn

The functional equation formula in the answer by Peter Humphries is for the Dirichlet eta function and corresponds to the second term in this sequence. This sequence corresponds to zeta function products over all the divisors.

Peter Humphries, What completes the Dirichlet generating function ζ(s+c-1) where c is a constant?, Math StackExchange.
Peter Humphries, What is the symmetric functional equation of the Dirichlet eta function?, Math StackExchange.

Cf. A000169, A014963, A120112, A230283, A230284.

Clear[s]; -Table[Limit[Zeta[s]*Total[MoebiusMu[Divisors[n]]*Divisors[n]^(1 - (s))]*Total[MoebiusMu[Divisors[n]]*Divisors[n]^(s)], s -> 1], {n, 1, 30}]; Exp[%]

a(n) = exp(lim_{s->1} zeta(s)*Sum_{d|n} mu(d)*d^(1 - s)*Sum_{d|n} mu(d)*d^(s)).
a(n) = A014963(n)^(A014963(n)-1), n > 1.
a(n) = A014963(n)^(-A120112(n)), n > 1.
a(prime(n)) = A000169(prime(n)).

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A133233 Triangle A133232 read by rows with an additional column T(n,0)=1 added to the left.

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A133232 Triangle T(n,k) read by rows with a minimum number of prime powers A100994 for which the least common multiple of T(n,1),..,T(n,n) is A003418(n).

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A120111 Bi-diagonal inverse matrix of A120108.

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A276048 Sequence associated with the functional equation of the Riemann zeta zero spectrum (see formulas).

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